A Spectral Gap Estimate and Applications

We consider the Schrödinger operator

$$ text{-} frac{d^{2}}{d x^{2}} + V {text{on an interval}}~~[a,b]~{text{with Dirichlet boundary conditions}},$$

where V is bounded from below and prove a lower bound on the first eigenvalue λ ₁ in terms of sublevel estimates: if w _V (y) = |{x ∈ [a, b] : V (x) ≤ y}|, then

$$lambda_{1} geq frac{1}{250} minlimits_{y > min V}{left( frac{1}{w_{V}(y)^{2}} + yright)}.$$

The result is sharp up to a universal constant if {x ∈ [a, b] : V(x) ≤ y} is an interval for the value of y solving the minimization problem. An immediate application is as follows: let ({Omega } subset mathbb {R}^{2}) be a convex domain and let (u:{Omega } rightarrow mathbb {R}) be the first eigenfunction of the Laplacian ? Δ on Ω with Dirichlet boundary conditions on ?Ω. We prove

$$| u |_{L^{infty}({Omega})} lesssim frac{1}{text{inrad}({Omega})} left( frac{text{inrad}({Omega})}{text{diam}({Omega})} right)^{1/6} |u|_{L^{2}({Omega})},$$

which answers a question of van den Berg in the special case of two dimensions.